Wordorigins.org

Keep your nose clean is an Americanism dating to 1887. From the New Orleans Lantern, 13 October of that year:

There’s worse fellows than you looking for it, and if you only keep your nose clean, we’ll let you have it.

It is almost certainly a metaphorical reference to a child maintaining proper hygiene in polite company, in particular children with runny noses. One is expected by be on good behavior and presentable just as a child is among adults.

(Sources: Oxford English Dictionary, 2nd Edition; Historical Dictionary of American Slang)